Did Cardano accept imaginary numbers as genuine 'numbers'?

No. He treated them as a computational device and did not regard them as genuine numbers. A real justification had to wait for Gauss.

Why are they called 'imaginary' numbers?

The term derives from the Latin imaginarius ('imagined'). In contrast to real numbers it once carried the nuance of 'numbers that do not exist', though today the name merely reflects this historical background.

Can cubic equations not be solved without imaginary numbers?

Even when a cubic has only real solutions, Cardano's formula passes through imaginary numbers. This is known as the casus irreducibilis (the irreducible case). There is an alternative method using trigonometric functions, but the approach via imaginary numbers is more general.

The Birth of Imaginary Numbers

A Brief History

Goal of this page

Learn when, why, and how "imaginary numbers" were discovered.

Prerequisites

A naive notion of cubic equations
An interest in history

Table of Contents

§1 Mathematicians of 16th-Century Italy
§2 Bombelli's "Wild" Idea
§3 The Road to Legitimate Mathematics
§4 Summary

§1 Mathematicians of 16th-Century Italy

In 16th-century Italy a heated mathematical dispute raged over how to solve cubic equations. Mathematicians such as Tartaglia, Cardano, and Bombelli competed to find a solution formula.

In 1545 Cardano published Ars Magna (The Great Art) and made public a formula for solving cubic equations (Cardano's formula). Yet something strange happened with this formula: even for cubic equations possessing only real solutions, a "square root of a negative number" could appear partway through the calculation.

$x^3 = 15x + 4$ via Cardano's formula:

$x = \sqrt[3]{2 + 11i} + \sqrt[3]{2 - 11i}$

imaginary numbers appear partway!

→ Bombelli simplifies:

$\sqrt[3]{2 \pm 11i} = 2 \pm i$

$\therefore\ x = (2 + i) + (2 - i) = 4$

passing through imaginary numbers to reach a real solution

Figure 1: The imaginary quantity $\sqrt[3]{2 \pm 11i}\,(= 2 \pm i)$ in Cardano's formula serves as a stepping stone to the real solution $x = 4$.

§2 Bombelli's "Wild" Idea

In 1572 Bombelli published Algebra and laid out a way to treat $\sqrt{-1}$ as a "symbol that obeys the rules of calculation." He called it "a wild idea," yet out of practical necessity he built up an algebra of imaginary numbers.

Example: $x^3 = 15x + 4$

Clearly $x = 4$ is a solution, but substituting into Cardano's formula produces $\sqrt{-121}$. Using Bombelli's rule $\sqrt{-121} = 11\sqrt{-1}$, one arrives at $x = 4$ by way of the cube roots.

§3 The Road to Legitimate Mathematics

In the 17th and 18th centuries, Descartes, Euler, and Gauss treated imaginary numbers systematically. Descartes left behind the somewhat dismissive name "imaginary" (suggesting a merely imagined, non-existent quantity). Euler introduced the symbol $i$, and in 1797 Gauss presented the complex plane, giving imaginary numbers a geometric justification.

Over some 250 years, imaginary numbers thus rose from a "wild" concept to a "legitimate number." Today they are at work in every corner of physics, engineering, and mathematics.

§4 Summary

Imaginary numbers arose inevitably from solving cubic equations
Cardano (1545) and Bombelli (1572) were the pioneers
Gauss's complex plane (1797) gave them a geometric justification
The name "imaginary" is a vestige of the prejudice of the time